We consider the problem of estimating the latent structure of a social network based on the observed information diffusion events, or {\it cascades}. Here for a given cascade, we only observe the times of infection for infected nodes but not the source of the infection. Most of the existing work on this problem has focused on estimating a diffusion matrix without any structural assumptions on it. In this paper, we propose a novel model based on the intuition that an information is more likely to propagate among two nodes if they are interested in similar topics which are also prominent in the information content. In particular, our model endows each node with an influence vector (which measures how authoritative the node is on each topic) and a receptivity vector (which measures how susceptible the node is for each topic). We show how this node-topic structure can be estimated from the observed cascades and prove an analytical upper bound on the estimation error. The estimated model can be used to build recommendation systems based on the receptivity vectors, as well as for marketing based on the influence vectors. Experiments on synthetic and real data demonstrate the improved performance and better interpretability of our model compared to existing state-of-the-art methods.

Traditional works on community detection from observations of information cascade assume that a single adjacency matrix parametrizes all the observed cascades. However, in reality the connection structure usually does not stay the same across cascades. For example, different people have different topics of interest, therefore the connection structure would depend on the information/topic content of the cascade. In this paper we consider the case where we observe a sequence of noisy adjacency matrices triggered by information/events with different topic distributions. We propose a novel latent model using the intuition that the connection is more likely to exist between two nodes if they are interested in similar topics, which are common with the information/event. Specifically, we endow each node two node-topic vectors: an influence vector that measures how much influential/authoritative they are on each topic; and a receptivity vector that measures how much receptive/susceptible they are to each topic. We show how these two node-topic structures can be estimated from observed adjacency matrices with theoretical guarantee, in cases where the topic distributions of the information/events are known, as well as when they are unknown. Extensive experiments on synthetic and real data demonstrate the effectiveness of our model.

We study the problem of recovery of matrices that are simultaneously low rank and row and/or column sparse. Such matrices appear in recent applications in cognitive neuroscience, imaging, computer vision, macroeconomics, and genetics. We propose a GDT (Gradient Descent with hard Thresholding) algorithm to efficiently recover matrices with such structure, by minimizing a bi-convex function over a nonconvex set of constraints. We show linear convergence of the iterates obtained by GDT to a region within statistical error of an optimal solution. As an application of our method, we consider multi-task learning problems and show that the statistical error rate obtained by GDT is near optimal compared to minimax rate. Experiments demonstrate competitive performance and much faster running speed compared to existing methods, on both simulations and real data sets.

Probabilistic graphical models have been widely used to model complex systems and aid scientific discoveries. As a result, there is a large body of literature focused on consistent model selection. However, scientists are often interested in understanding uncertainty associated with the estimated parameters, which current literature has not addressed thoroughly. In this paper, we propose a novel estimator for edge parameters for pairwise graphical models based on Hyv\"arinen scoring rule. Hyv\"arinen scoring rule is especially useful in cases where the normalizing constant cannot be obtained efficiently in a closed form. We prove that the estimator is $\sqrt{n}$-consistent and asymptotically Normal. This result allows us to construct confidence intervals for edge parameters, as well as, hypothesis tests. We establish our results under conditions that are typically assumed in the literature for consistent estimation. However, we do not require that the estimator consistently recovers the graph structure. In particular, we prove that the asymptotic distribution of the estimator is robust to model selection mistakes and uniformly valid for a large number of data-generating processes. We illustrate validity of our estimator through extensive simulation studies.